Fourier Integrals in Classical Analysis

✍ Scribed by Christopher D. Sogge

• Publisher: Cambridge University Press : Cambridge University Press
• Year: 2017
• Tongue: English
• Leaves: 349
• Series: Cambridge tracts in mathematics 210
• Edition: 2ed.
• Category: Library

✦ Synopsis

This advanced monograph is concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat-Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff

✦ Table of Contents

Content: Cover
Series page
Title page
Copyright information
Dedication
Table of contents
Preface to the Second Edition
Preface to the First Edition
0 Background
0.1 Fourier Transform
0.2 Basic Real Variable Theory
0.3 Fractional Integration and Sobolev Embedding Theorems
0.4 Wave Front Sets and the Cotangent Bundle
0.5 Oscillatory Integrals
Notes
1 Stationary Phase
1.1 Stationary Phase Estimates
The One-Dimensional Case
Stationary Phase in Higher Dimensions
1.2 Fourier Transform of Surface-carried Measures
Notes
2 Non-homogeneous Oscillatory Integral Operators 2.1 Non-degenerate Oscillatory Integral Operators2.2 Oscillatory Integral Operators Related to the Restriction Theorem
2.3 Riesz Means in Rn
2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2
Notes
3 Pseudo-differential Operators
3.1 Some Basics
3.2 Equivalence of Phase Functions
3.3 Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds
Notes
4 The Half-wave Operator and Functions of Pseudo-differential Operators
4.1 The Half-wave Operator
4.2 The Sharp Weyl Formula
4.3 Smooth Functions of Pseudo-differential Operators
Notes 5 Lp Estimates of Eigenfunctions5.1 The Discrete L2 Restriction Theorem
Application: Unique Continuation for the Laplacian
5.2 Estimates for Riesz Means
5.3 More General Multiplier Theorems
Notes
6 Fourier Integral Operators
6.1 Lagrangian Distributions
6.2 Regularity Properties
Sharpness of Results
6.3 Spherical Maximal Theorems: Take 1
Notes
7 Propagation of Singularities and Refined Estimates
7.1 Wave Front Sets Redux
7.2 Propagation of Singularities
7.3 Improved Sup-norm Estimates of Eigenfunctions
7.4 Improved Spectral Asymptotics
Notes 8 Local Smoothing of Fourier Integral Operators8.1 Local Smoothing in Two Dimensions and Variable Coefficient Nikodym Maximal Theorems
Orthogonality Arguments in Two Dimensions
Variable Coefficient Nikodym Maximal Functions
8.2 Local Smoothing in Higher Dimensions
Orthogonality Arguments in Higher Dimensions
8.3 Spherical Maximal Theorems Revisited
Notes
9 Kakeya-type Maximal Operators
9.1 The Kakeya Maximal Operator and the Kakeya Problem
9.2 Universal Bounds for Kakeya-type Maximal Operators
9.3 Negative Results in Curved Spaces Negative Results for Oscillatory Integrals Arising in Curved Spaces9.4 Wolff's Bounds for Kakeya-type Maximal Operators
Auxiliary Maximal Function Bounds
Wolff's Bounds for Nikodym Maximal Functions in Rn
9.5 The Fourier Restriction Problem and the Kakeya Problem
Notes
Appendix Lagrangian Subspaces of T∗Rn
References
Index of Notation
Index